Sorting in Heterogeneous Contests

This paper establishes the efficient sorting of contests when the actions of workers are not observable to firms and the workers are heterogeneous by utilizing the assumption that the productivities of each type of worker vary with the firms. The previous result of ‘nonexistence’ of efficient sorting of contests is shown due to the strict use of the risk-neutral and separable utility functions. Efficient sorting is possible because both the probability of winning and the level of prizes determine the expected return in contests, which move in the opposite directions. [610]     

Contests with rank-order spillovers

This paper presents a unified framework for characterizing symmetric equilibrium in simultaneous move, two-player, rank-order contests with complete information, in which each player??s strategy generates direct or indirect affine ??spillover?? effects that depend on the rank-order of her decision variable. These effects arise in natural interpretations of a number of important economic environments, as well as in classic contests adapted to recent experimental and behavioral models where individuals exhibit inequality aversion or regret. We provide the closed-form solution for the symmetric Nash equilibria of this class of games, and show how it can be used to directly solve for equilibrium behavior in auctions, pricing games, tournaments, R&D races, models of litigation, and a host of other contests.     

Contests with size effects

In this paper we analyze the structure of contest equilibria with a variable number of individuals. First we analyze a situation where the total prize depends on the number of agents and where every single agent faces opportunity costs of investing in the contest. Second we analyze a situation where the agents face a trade-off between productive and appropriative investments. Here, the number of agents may also influence the productivity of productive investments. It turns out that both types of contests may lead to opposing results concerning the optimal number of individuals depending on the strength of size effects. Whereas in the former case individual utility is u-shaped when the number of agents increases, the opposite holds true for the latter case. We discuss the implications of our findings for the case of anarchic societies and market competition.     

Contests with limited resources

Many interesting phenomena (electoral competition, R&D races, lobbying) are instances of multiple simultaneous contests with unconditional commitment of limited resources. Specifically, the following game is analyzed. Two players compete in a number of simultaneous contests. The players have limited resources (budgets) and must decide how to allocate these to the different contests. In each contest the player who expends more resources than his adversary wins a corresponding prize. Mixed-strategy equilibria are characterized in the case of identical values and budgets and the connections with the classical Blotto game are analyzed.     

Contests with Multi-tasking*

The standard contest model in which participants compete in a single dimension is well understood and documented. Multi‐dimension extensions are possible but are liable to increase the complexity of the contest structure, mitigating one of its main advantages: simplicity. In this paper we propose an extension in which competition ensues in several dimensions, and a competitor that wins a certain number of these is awarded a prize. The amount of information needed to run the contest is hence limited to the number of dimensions won by each player. We look at the design of this contest from the point of view of maximising effort in the contest (per dimension and totally), and show that there will be a tendency to run small contests with few dimensions. The standard Tullock model and its results are encompassed by our framework.     

Orchestrating Rent Seeking Contests

Contests have different, sometimes quite complex, organisational structures. In particular, while most of the existing literature focusses on simultaneous contests, multistage contests are also quite frequently encountered. This paper seeks to provide a rationale for the latter by endogenising the choice of a contest structure, which is made by an organiser of a contest interested in maximising the efforts expended by the contenders.     

Repeated Contests with Asymmetric Information

The same contestants often meet repeatedly in contests. Behavior in a contest potentially provides information with regard to one's type and can therefore influence the behavior of the opponents in later contests. This paper shows that if effort is observable, this can induce a ratchet effect in contests: high ability contestants sometimes put in little effort in an early round in order to make the opponents believe that they are of little ability. The effect reduces overall effort and increases equilibrium utility of the contestants when compared with two unrelated one-shot contests. It does, however, also introduce an allocative inefficiency since sometimes a contestant with a low valuation wins. The model assumes an imperfectly discriminating contest. In an extension I show that, qualitatively, results are similar in a perfectly discriminating contest (all pay auction).     

Marginal Subsidies in Tullock Contests

In a general Tullock contest, we examine a situation where a limited resource can be used to provide marginal subsidies to either player (weak or strong), or to increase the prize directly. We show that to maximize total effort, subsidizing the weak/strong player is preferred when the contest is sufficiently accurate/inaccurate. This result generalizes to n‐player lottery contests. In a lottery contest (Tullock contest with ), we derive the optimal scheme for a full range of resource: when the resource is small, it is optimal to only subsidize the weak player; when it is large, both players should be subsidized simultaneously.     

Difference‐Form Persuasion Contests

We explore the equilibrium properties of two types of “difference‐form” persuasion contest functions derived in Skaperdas and Vaidya in which contestants spend resources to persuade an audience. We find that both types of functions generate interior pure strategy Nash equilibria unlike Baik and Che and Gale with characteristics different to existing literature. For one type of function, we find that the reaction function of each player is “flat” and nonresponsive to the level of resources devoted by the rival so that the “preemption effect” as defined by Che and Gale is absent. Further, the equilibrium is invariant to the sequencing of moves. For the second type of function, which applies when there is asymmetry among contestants with regard to the quality of evidence, we find that the reaction functions of the stronger and weaker players have gradients with opposite signs relative to Dixit and therefore their incentive to precommit expenditures in a sequential move game is also different. For both types of functions, the extent of rent dissipation is partial. From the equilibrium analysis, we are also able to establish the potential effects of some specific factors affecting persuasion such as evidence potency, the degree of truth, and bias on aggregate resource expenditures and welfare.     

Takeover Contests, Toeholds and Deterrence

We consider a setting in which two potential buyers, one with a prior toehold and one without, compete in a takeover modeled as an ascending auction with participating costs. The toeholder is more aggressive during the takeover process because she is also a seller of her own shares. The non-toeholder anticipates this extra-aggressiveness of the toeholder. Thus, he is deterred from participating unless he has a high valuation for the target company. This leads to large inefficiency losses. For many configurations, expected target returns are first increasing then decreasing in the size of the toehold.     

Dynamic Difference‐Form Contests

This study presents a multistage contest designed by an organizer who chooses the (indivisible) winner prize together with the ranking scheme. When the organizer's objective is to maximize efforts, the optimal ranking scheme that selects the prize recipient coincides with a version of the piecewise linear difference‐form success function postulated by Che and Gale (2000). Even if the organizer's objective is not necessarily to maximize agents' effort, a noisy ranking of the difference‐form remains optimal.     

Contests with size effects through costs

Eggert and Kolmar [Eggert, W., Kolmar, M., in press. Contests with size effects. European Journal of Political Economy.] examine the effects of group size on individual and aggregate payoffs in contests where prizes increase in the number of players. In a reconsideration of their paper, this note examines contests in which the unit cost of effort decreases in the number of players. In contests for fixed prizes, an increase in the number of players decreases individual and aggregate payoffs. In contests in which players face a trade-off between appropriative and productive investments, an increase in the number of players decreases individual payoffs while increasing aggregate payoffs.     

Imitation and Experimentation in Changing Contests

This paper analyzes the equilibrium play of individuals that are randomly matched to play a contest where the dominant action changes over time. Under myopic decision making, players adopt imitation strategies similar to those observed in evolutionary models with sampling from past play in the population. If the players are patient, equilibrium strategies display elements of experimentation in addition to imitation. If the changes in the environment are infrequent enough, these strategies succeed in coordinating almost all of the players on the dominant action almost all of the time. The myopic rules, on the other hand, result in miscoordination for a positive fraction of time. Journal of Economic Literature Classification Numbers: C73, D83, D84.     

Contests with spying: A comment

The subject of the following comment is a paper by Baik and Shogren which has been published in this journal (Vol. 11, No. 3 (1995) pp. 441–451). In their game-theoretic model, Baik and Shogren connect the informational states of the agents in a way which leads to inconsistencies.     

Dynamic Multi‐Activity Contests*

We develop a model of dynamic multi‐activity contests. Players simultaneously choose efforts in long‐run activities, observe each other's efforts in these activities, and then simultaneously choose efforts in short‐run activities. A player's long‐run and short‐run efforts complement each other in determining the player's probability of winning. We compare the outcomes of this two‐stage model to those of the corresponding model in which players choose efforts in all activities simultaneously. Interestingly, effort expenditures are always lower in the sequential multi‐activity contest than in the simultaneous multi‐activity contest. The implications of this result for the organization of military, litigation, innovation, academic, and sporting contests are highlighted.     

Exit from Rent‐Seeking Contests

The object of this paper is to investigate the long‐run behaviour of rent‐seekers in a situation where they incur negative expected payoffs due to increasing returns to rent‐seeking expenditures. To this end, we embed the one‐shot rent‐seeking game presented by Tullock (1980) in the war‐of‐attrition framework. In this multi‐period setting, each player not only determines his or her rent‐seeking expenditure but also chooses a mixed strategy on whether to stay in or exit from rent‐seeking competition in each period. JEL Classification Numbers: D72, C72, L12     

Effort and Performance in Public Policy Contests

Government intervention often gives rise to contests in which the possible “prizes” are determined by the status quo and some new public policy proposal. In this paper we study a general class of such two‐player public policy contests and examine the effect of a change in the proposed policy, a change that may affect the payoffs of the two contestants, on their effort and performance. Our results extend the existing comparative statics studies that focus, in symmetric contests, on the effect of a change in the value of the prize or, in asymmetric contests, on the effect of one contestant's valuation of the prize. Our results hinge on a fundamental equation that specifies the equilibrium relationship between the strategic own‐stake effect and the strategic rival's‐stake effect. This fundamental equation clarifies the role of the three possible types of ability and stake asymmetry in determining the effect of payoff variations on the efforts and performance of the contestants.     

Contests, NGOs, and Decentralizing Aid

International donors usually have particular goals they want to achieve with their foreign aid, e.g., poverty alleviation. In the international aid story lobbying by potential recipient groups attempting to capture the donor's support plays a potentially important role for non‐governmental organizations. We model this situation as a hierarchical contest and compare the implications of a centralized allocation process with a decentralized allocation process with non‐governmental organizations as intermediaries.     

Contests with a stochastic number of players

We study Tullock's (1980) n-player contest when each player has an independent probability 0 < p 1 of participating. A unique symmetric equilibrium is found for any n and p and its properties are analyzed. In particular, we show that for a fixed n > 2 individual equilibrium spending as a function of p is single-peaked and satisfies a single-crossing property for any two different numbers of potential players. However, total equilibrium spending is monotonically increasing in p and n. We also demonstrate that ex-post over-dissipation is a feature of the pure-strategy equilibrium in our model. It turns out that if the contest designer can strategically decide whether to reveal the actual number of participating players or not, then the actual number of participants is always revealed.